When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure Theory.

One application I know of for this theorem is to recurrence relations. One shows that a recurrence relation like $a_{n+1} = \sqrt{2 + a_n}, a_0 = 0$ is non-decreasing and bounded, from which one concludes that it has a limit as $n$ approaches $\infty$.

Are there any other applications of the theorem?

I ask because I'm curious about the possibility of systematically "constructivising" applications of this theorem. Obviously in general this isn't possible. But surely there are lots of situations in which it may be possible.

I tried constructivising Herschfeld's Convergence Theorem, I believe successfully. The resulting argument is a great deal less straightforward than Herschfeld's original one.

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There was a slight misunderstanding in the comments. The Cauchy completeness of the real numbers does not imply the Monotone Convergence Theorem, unless one assumes the Law of Excluded Middle.